TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 10, October 1998, Pages 4167–4193 S 0002-9947(98)02125-4

THE STRUCTURE OF INDECOMPOSABLE INJECTIVES IN GENERIC REPRESENTATION THEORY

GEOFFREY M. L. POWELL

Abstract. This paper considers the structure of the injective objects I_{V_{n}} in the category F of functors between 𝔽_{2}-vector spaces. A co-Weyl object J_{λ} is defined, for each simple functor F_{λ} in F. A functor is defined to be J-good if it admits a finite filtration of which the quotients are co-Weyl objects. Properties of J-good functors are considered and it is shown that the indecomposable injectives in F are J-good. A finiteness result for proper sub-functors of co-Weyl objects is proven, using the polynomial filtration of the shift functor Δ̃ : F → F. This research is motivated by the Artinian conjecture due to Kuhn, Lannes and Schwartz.

1. Introduction

The purpose of this paper is to study the structure of the indecomposable injective objects in the category F of functors E_{f} → E, where E is the category of 𝔽-vector spaces and E_{f} the full sub-category of finite dimensional spaces. The field 𝔽 is taken to be a finite field and, for the purposes of this introduction and for some of the main results, this is taken to be 𝔽_{2}, the prime field with two elements. Kuhn has termed F the category of generic representations in his sequence of papers [K1, K2, K3]. The special interest of this category of functors in the finite field case follows from the link with the category of unstable modules over the Steenrod algebra. A review of the category F is presented in Section 2, to which the reader is referred for any unexplained terminology.

The paper shows that a high degree of understanding of the structure of the indecomposable injective functors in F is obtained by concentrating on certain smaller building blocks, which the author has named co-Weyl objects, by analogy with the theory of highest weight categories, introduced by Cline, Parshall and Scott in [CPS]. It is readily seen that F does not have the structure of a highest weight category; however, many of the features of such a category may be recovered by new techniques.

The simple functors in F correspond to simple GL_{n}(𝔽) modules, as n varies. A co-Weyl object is associated to each simple functor: if F_{λ} is a simple functor indexed by λ, then the corresponding co-Weyl object is written as J_{λ}. This is the largest analytic functor such that J_{λ} has simple socle F_{λ} and J_{λ}(𝔽^{n}) = F_{λ}(𝔽^{n}), where n is the smallest integer such that F_{λ}(𝔽^{n}) ≠ 0.

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors December 5, 1996.

1991 Mathematics Subject Classification. Primary 18G05, 20G40, 55S10.

The author was supported by a Royal Society (GB) ESEP fellowship at the Institut Galilée, Université de Paris-Nord, France during the preparation of this work. The final version was prepared whilst the author was a visitor at the University of Virginia, Charlottesville.

A functor F ∈ F is said to be J-good if it has a finite filtration of which the quotients are co-Weyl objects. The justification for interest in such functors is provided by the first important result of this paper, which is proved in Section 4; it may be regarded as saying that the co-Weyl objects are the building blocks for the category F.

Theorem 1. The indecomposable injective functors in F are J-good.

The proof of this result deserves some comment; recall (see Section 2) that I_{V} denotes a standard injective in F, that M_{n} is the category of End(𝔽^{n})-modules and that GL_{n} denotes the category of Aut(𝔽^{n})-modules, which may be considered as a full sub-category of M_{n}. The evaluation functor E_{n} : F → M_{n}, ( F ↦ F (𝔽^{n})) admits a right adjoint R_{n} : M_{n} → F. The functor R_{n} is not exact; however, it is shown in Section 2 that the restriction of R_{n} to GL_{n} is exact. This is the key step in the proof that the functors I_{V} are J-good, since it may be shown that J_{λ} ≅ R_{n}(S_{λ}), where n is the least integer such that F_{λ}(𝔽^{n}) is non-zero and S_{λ} is the simple GL_{n}-module F_{λ}(𝔽^{n}).

The proof of the theorem then requires the use of certain properties of the category of J-good functors, in particular that a direct summand of a J-good functor is again J-good. The proof of this result requires an estimate of the size of co-Weyl functors. This is achieved by using the next key ingredient of the paper, derived from the study of the polynomial filtration of the shift functor Δ̃ : F → F, which was introduced in [P]. This gives a collection of functors ˜∇_{n}, which are linked by natural surjections ˜∇_{n} →→ ˜∇_{n+1}. These functors have some good properties; in particular, they preserve injections and surjections, although they are not exact. This paper shows that the concept of ˜∇_{n}-nilpotence is important: a functor F is said to be ˜∇_{n}-nilpotent if there exists an integer k such that ( ˜∇_{n})^{k}F = 0. This provides the required estimate of the size of a co-Weyl object: if J_{λ} = R_{n}(S_{λ}), where S_{λ} is a simple GL_{n}-module, then J_{λ} is not ˜∇_{n}-nilpotent, whereas ˜∇_{n+1}J_{λ} = 0.

The theorem may be used to show that the Cartan invariants dim Hom_{F}(I_{λ}, I_{μ}) in the category F determine the multiplicity of co-Weyl functors in J-good filtrations of the indecomposable injectives. In particular, this gives a proof of the following result:

Corollary 2. The Cartan matrix for 𝔽[M_{n}] determines the Cartan matrix for 𝔽[GL_{n}].

In addition, the author has determined an homological characterization of Jgood functors, based on the fact that the indecomposable injectives in F are Jgood. This was inspired by results in algebraic group theory related to the notion of good filtration dimensions (see for example [FP] or [J]). In the statement of the result below (and throughout the paper) V_{d} represents the 𝔽-vector space of dimension d and I_{V_{d}} is the associated injective.

Theorem 3. An analytic functor F with finite socle is J-good if and only if there is an integer d such that F has an injective resolution 0 → F → I^{•}, in which each I^{k} is a finite direct sum ⨁ I_{V_{d}}.

This result is proved in Section 5 and plays a pivotal role in the paper; it implies immediately that the tensor product of J-good functors is J-good and also that the difference functor of a J-good functor is J-good. This shows that ˜∇_{n}J_{λ} occurs as a quotient of a J-good functor, namely ΔJ_{λ}. This is a major input into Proposition 6.0.4, which shows that ˜∇_{n}J_{λ} ≅ J_{λ} (except in one special case), when n is the least integer such that J_{λ}(𝔽^{n}) ≠ 0.

This makes the following result striking; it appears as Theorem 6.0.1. The author regards this as the most important result of the paper, since it yields considerable information about the structure of the injectives in F.

Theorem 4. Suppose that J_{λ} is a co-Weyl functor induced from a simple GL_{n}- module. If F ↪ J_{λ} is a proper sub-functor, then F is ˜∇_{n}-nilpotent.

The author has termed this result the Simplicity Theorem, since it implies that J_{λ} is a simple object in the abelian category F_{ω}/ ˜∇_{n} − N il of analytic functors localized away from the full sub-category of ˜∇_{n}-nilpotent functors. In a future paper, the author will exploit this theorem to show how this implies certain results about extension groups in F, which have interesting corollaries for the category of unstable modules over the Steenrod algebra.

Section 7 briefly discusses the category B^{E_{f}} of Boolean algebra valued functors, motivated by the fact that these correspond (under the work of [HLS]) to unstable algebras over the Steenrod algebra. It is shown that:

Theorem 5. If K ∈ B^{E_{f}} has finite socle, then K is J-good as a functor in F.

This result shows that many of the functors in F which arise from the cohomology of a space are J-good; the hypothesis essentially restricts us to the study of algebras of finite transcendence degree, which take finite-dimensional values.

1.1. Remarks on the odd prime case. Most of the results discussed above may be extended to the odd primary case, namely the case where 𝔽 = 𝔽_{p}, where p is an odd prime. The essential change is that now one uses the fact that ˜∇_{(p−1)n}I_{V_{n}} ≅ I_{V_{n}} and ˜∇_{(p−1)n+1}I_{V_{n}} = 0.

1.2. Concerning the Artinian conjecture. In this section, the implication of the results of this paper for the study of the conjecture of the title is explained; again it is supposed that 𝔽 is the field 𝔽_{2}. Recall that F is an abelian category; an object F ∈ F is said to be Artinian if every descending sequence of sub-functors of F stabilizes. The following has been conjectured; it appears in print in [K1] and is attributed largely to Lionel Schwartz, in conjunction with Nick Kuhn and Jean Lannes.

The Artinian conjecture: The functors I_{V_{n}} are Artinian.

An affirmative resolution of this conjecture would have the consequence that an analytic functor F with finite socle would have an injective resolution 0 → F → I^{•}, in which each I^{k} has finite socle. The calculation of the extension groups Ext_{F}(−, −) is related to the study of Mac Lane cohomology with coefficients in twisted bifunctors of coefficients (see [FLS]) and with topological Hochschild cohomology; therefore the knowledge of such a finiteness result for injective resolutions would imply finiteness results for these theories, at least when the underlying ring is a finite field.

A stronger formulation of the conjecture may be given, using the following terminology. A functor F is said to be simple Artinian of type 0 if it is a simple functor; F is Artinian of type n if it has a finite filtration of which the sub-quotients are simple Artinian of type n. Finally, F is simple Artinian of type n if every proper sub-functor of F is Artinian of type n − 1.

The strong Artinian conjecture: The functors I_{V_{n}} are Artinian of type n.

This conjecture is known to be true for the functors I_{𝔽}, for any finite field 𝔽. For 𝔽 = 𝔽_{2}, the only other case known is I_{𝔽^{2}}, which was shown by the author to be Artinian of type two, in [PS]. The results of this paper, together with the methods of [PS] lead us to propose:

The very strong Artinian conjecture: If S_{λ} is a simple GL_{n}-module, then the associated co-Weyl object J_{λ} = R_{n}S_{λ} is simple Artinian of type n.

It is not difficult to deduce that the very strong conjecture would imply the strong conjecture. The simplicity theorem of this paper implies the following reduction:

Proposition 1.2.1. The very strong Artinian conjecture holds for all J_{λ} induced from simple GL_{n}-modules if and only if the strong Artinian conjecture holds for I_{V_{n−1}} and every sub-functor of I_{V_{n}} which is ˜∇_{n}-nilpotent is Artinian of type n − 1.

This is of interest, since the largest sub-functor F of I_{V_{n}} with ( ˜∇_{n})^{t}F = 0 is the kernel of an explicit map

I_{V_{n}} → I_{V_{n+t}},

which is induced by an element 𝔽[hom(V_{n}, V_{n+t})] which is described in [P]. It follows that the very strong Artinian conjecture may be reduced to studying certain explicit ‘small’ sub-functors of I_{V_{n}}

2. The category F

2.1. Review of the category F. The category F is abelian, with structure induced from the abelian category E of 𝔽-vector spaces. In particular, a sequence F → G → H of functors is short exact if and only if it is short exact when evaluated on any finite dimensional vector space. The usual definitions of simple object and composition series are taken. The classification of the simple objects is reviewed in Section 2.3. Good references for this material are the papers [K1, K2, K3], together with the first half of the book by Lionel Schwartz [S].

The category F is equipped with a duality functor, D : F → F^{op}, which generalizes the notion of transpose duality in representation theory. This is defined by DF (V) := F (V^{∗})^{∗}, where the ∗ denotes the vector space dual. If F takes finite dimensional values, then it follows that there is a natural isomorphism DDF ≅ F.

A functor F ∈ F is said to be finite if it has a finite composition series and is said to be analytic if it is the colimit of its finite sub-objects. The full sub-category of F with objects which are the analytic functors is denoted by F_{ω} and is termed the category of analytic functors.

The category F has enough injectives and projectives; the standard projective P_{V} is defined by P_{V} (𝔽^{n}) = 𝔽[hom(V, 𝔽^{n})], where V is a finite dimensional vector space. Up to isomorphism, this is determined by the representing property:

Hom_{F}(P_{V} , F) ≅ F (V),

naturally in the functor F. The standard injective I_{V} may be defined as the dual DP_{V} of the projective. Explicitly, this means that:

I_{V} (𝔽^{n}) = 𝔽^{hom(V,𝔽^{n} ^{∗})}.

Again, up to isomorphism, this is determined by a co-representing property, which is dual to that given above. From this discussion it is clear that:

Hom_{F}(I_{V} , I_{W}) ≅ 𝔽[hom(V, W)],

so that one may regard I_{−} as a covariant functor from E_{f} to F.

The injective functors I_{V} are analytic, whereas P_{V} is not analytic in general. In fact, P_{V} splits as 𝔽 ⊕ P̂_{V} and P̂_{V} has no finite sub-objects.

The category F is equipped with the shift functor Δ̃ : F → F, which is defined on objects by Δ̃F (V) = F (V ⊕ 𝔽). This functor is right adjoint to the functor − ⊗ P_{𝔽}. The functor Δ̃ splits naturally as Δ̃ ≅ Δ ⊕ id, where the splitting is given by the natural split injection F (V) ↪ F (V ⊕ 𝔽); the functor Δ is known as the difference functor.

An important concept in F is that of the polynomial degree of a functor, which has been studied since the work of Eilenberg and Mac Lane [EM]. The definition of a polynomial functor taken here uses the difference functor. Namely, a functor F is said to be polynomial of degree ≤ d if Δ^{d+1}F = 0. The full sub-category of polynomial functors of degree ≤ d is denoted by F_{d}. The inclusion F_{d} ↪ F admits a left adjoint q_{d} : F → F_{d} and a right adjoint p_{d} : F → F_{d}. These are related by a natural isomorphism Dq_{d}F ≅ p_{d}DF.

The shift functor Δ̃ admits a polynomial filtration, which was introduced in [P]. This may be illustrated by observing that there is an isomorphism:

Δ̃I_{V} ≅ 𝔽^{V^{∗}} ⊗ I_{V} .

Here 𝔽^{V^{∗}} may be regarded as the free 𝔽-Boolean algebra on a vector space basis of V and is a constant functor. This admits the usual notion of a polynomial filtration, which induces a filtration of Δ̃I_{V}.

Formally, one may define the functor [p_{k}Δ̃] to be the right adjoint to the functor − ⊗ q_{k}P_{𝔽}; by convention take [p_{−1}Δ̃] = 0. It is useful to study the quotients ˜∇_{s} for s ≥ 0 and sub-quotients ∇_{s} defined below:

Definition 2.1.1. Suppose s ≥ 0.

1. ˜∇_{s} := Δ̃/[p_{s−1}Δ̃].

2. ∇_{s} := [p_{s}Δ̃]/[p_{s−1}Δ̃]. □

The functors ˜∇_{s} : F → F are the natural functors to be considering, although it can be useful in applications to use the functors ∇_{s}. One should observe that ˜∇_{1} identifies with the difference functor Δ; moreover, there are natural surjections ˜∇_{s} →→ ˜∇_{s+1}, for each s. The essential properties of these functors are summarized below:

Theorem 2.1.2. [P]

1. The functors ˜∇_{s} preserve injections, surjections and direct sums.

2. If F is an analytic functor with finite socle then ˜∇_{s}F is an analytic functor with finite socle.

3. The functor ˜∇_{s}I_{V} is injective.

4. If 𝔽 = 𝔽_{2}, then ∇_{s}I_{V} ≅ Λ^{s}(V) ⊗ I_{V}; if g : V → W is a map of vector spaces which induces [g] : I_{V} → I_{W}, then ∇_{s}g identifies with Λ^{s}(g) ⊗ [g] : Λ^{s}(g) ⊗ I_{V} → Λ^{s}(W) ⊗ I_{W}.

2.2. Recollements in F. Throughout this paper, M_{n} denotes the category of left M_{n} = End(𝔽^{n})-modules and GL_{n} is the category of left GL_{n}-modules, where GL_{n} is the group Aut(𝔽^{n}). The canonical inclusion Aut(𝔽^{n}) ⊂ End(𝔽^{n}) induces an inclusion 𝔽[GL_{n}] ↪ 𝔽[M_{n}], which is split by the ring map 𝔽[M_{n}] → 𝔽[GL_{n}] which sends singular endomorphisms to zero and automorphisms to themselves.

Kuhn has stressed in [K2] that a fundamental tool for studying the link between the representation theory of the above rings is the recollement diagram:

GL_{n} →^{i_{n}} M_{n} ⟵^{l_{n−1}} ; ⟶^{e_{n−1}} ; ⟵^{r_{n−1}} M_{n−1},

in which e_{n−1} : M_{n} → M_{n−1} is induced by a projection e_{n−1} ∈ End(V_{n}) onto 𝔽^{n−1}, via M ↦ e_{n−1}M. The functor l_{n−1} is the left adjoint to e_{n−1} and r_{n−1} is the right adjoint to e_{n−1}. These functors are given on objects N ∈ M_{n−1} by

l_{n−1}N := 𝔽[hom(𝔽^{n−1}, 𝔽^{n})] ⊗_{M_{n−1}} N,
r_{n−1}N := Hom_{M_{n−1}}(𝔽[hom(𝔽^{n}, 𝔽^{n−1})], N).

There are related recollement diagrams for the category F; write E_{n} : F → M_{n} for the evaluation functor F ↦ F (𝔽^{n}) and F^{n+1} for the kernel of E_{n}, which is the full sub-category of functors F with F (𝔽^{n}) = 0. There is a recollement diagram:

F^{n+1} → F ⟵^{L_{n}} ; ⟶^{E_{n}} ; ⟵^{R_{n}} M_{n},

where L_{n} and R_{n} are given by

L_{n}M := P_{V_{n}} ⊗_{M_{n}} M,
R_{n}M := Hom_{M_{n}}(𝔽[hom(−, V_{n})], M).

These functors are related via the duality functor, restricted to a functor D : M_{n} → M_{n}^{op}, which identifies with transpose duality:

D L_{n}M ≅ R_{n} DM.

The right adjoint to the inclusion F^{n+1} = Ker E_{n} ↪ F is also used throughout the paper; this is denoted by k_{n} : F → F^{n+1} and is defined on objects by k_{n}(F) := ker {F ↦ R_{n}E_{n}F }, where the map is the unit of the (E_{n}, R_{n})-adjunction.

The functor E_{n} : F → M_{n} restricts to the category F^{n} to give a functor which factorizes as F^{n} → GL_{n} →^{i_{n}} M_{n}, so that E_{n}|_{F^{n}} may be regarded as a functor E_{n} : F^{n} → GL_{n}, which fits into the restricted recollement diagram:

F^{n+1} → F^{n} ⟵^{L_{n}} ; ⟶^{E_{n}} ; ⟵^{R_{n}} GL_{n}.

The important fact about the above functors is given by

Proposition 2.2.1. The functors R_{n}, L_{n} : GL_{n} → F^{n} are exact.

Proof. Using the duality relation between L_{n} and R_{n}, it suffices to prove that the functor L_{n} : GL_{n} → F^{n} is exact. Recall that a GL_{n}-module is regarded as an object in M_{n} by using the functor i_{n} : GL_{n} → M_{n}, so that M is regarded as an 𝔽[End V_{n}] module via the canonical map 𝔽[End V_{n}] → 𝔽[GL_{n}]. Also, by definition, L_{n}M is isomorphic to the functor 𝔽[hom(V_{n}, −)] ⊗_{End V_{n}} M.

Since singular endomorphisms act trivially upon M, this identifies with the functor 𝔽[inj(V_{n}, −)] ⊗_{GL_{n}} M, where inj(V_{n}, W) denotes the set of injections from V_{n} to W and 𝔽[inj(V_{n}, −)] is given the structure of a functor by regarding it as a quotient of 𝔽[hom(V_{n}, −)]. Now, as a right GL_{n}-set, inj(V_{n}, W) identifies as Gr_{n}(W) × GL_{n}, the free GL_{n}-set, where Gr_{n}(W) is the Grassmannian of ndimensional planes in W. Hence, 𝔽[inj(V_{n}, W)] is free as a right GL_{n}-module, namely 𝔽[inj(V_{n}, W)] ≅ 𝔽[Gr_{n}(W)] ⊗ 𝔽[GL_{n}]. (Note that this is not intended to be an isomorphism of End(W)-modules.)

In particular, this gives that

L_{n}M(W) ≅ 𝔽[inj(V_{n}, W)]⊗_{GL_{n}M} ≅ 𝔽[Gr_{n}(W)] ⊗ M,

as vector spaces. The functor 𝔽[Gr_{n}(W)] ⊗ − is exact for each W, as a functor GL_{n} → E_{f}, so that the functor L_{n} is exact, as required, since exactness is only checked at the level of vector spaces.

Remark 2.2.2. It must be stressed that the functors R_{n}, L_{n}; M_{n} → F are not exact. The simplest example of this is given by taking 𝔽 = 𝔽_{2} and studying the M_{2}-module S^{2}(𝔽^{2}), where S^{2} is the second symmetric power. This module decomposes via the short exact sequence Λ^{2}(𝔽^{2}) → S^{2}(𝔽^{2}) → Λ^{1}(𝔽^{2}), which is a non-trivial extension.

It is not difficult to show that R_{2}(S^{2}(𝔽^{2})) ≅ \overline{I}, where I_{𝔽} ≅ 𝔽 ⊕ \overline{I} is the complete direct sum decomposition of I_{𝔽}. Moreover, R_{2}(Λ^{1}(𝔽^{2})) ≅ Λ^{1}. However, R_{2}(Λ^{2}(𝔽^{2})) is not isomorphic to \overline{I}/Λ^{1}, as is shown in Example 3.1.2. □

2.3. Review of the simple functors and indecomposable injectives. This section recalls briefly the representation theory of GL_{n} which underlies this paper, together with its connection with the category F. A p-regular partition of length s is taken to be a sequence of integers λ = (λ_{1} ≥ λ_{2} ≥ … ≥ λ_{s} > 0), where λ_{i} > λ_{i+p}, wherever this makes sense. If 𝔽 is a prime field of characteristic p, the simple GL_{n}-modules are indexed by the p-regular partitions with λ_{1} = n. The simple module S_{λ} indexed by λ is the unique top composition factor of the Weyl module W_{λ} ⊂ Λ^{λ} = Λ^{λ_{1}} ⊗ … ⊗ Λ^{λ_{s}}, where Λ^{k} denotes the k^{th} exterior power of the natural representation of GL_{n}.

The simple functors in F are indexed over all p-regular partitions, λ; if λ_{1} = n, then the simple functor F_{λ} is defined as the image of the natural map L_{n} S_{λ} → R_{n} S_{λ}, which is adjoint to the inverse of the unit of the (L_{n}, E_{n})-adjunction (which is an isomorphism, since this is a recollement situtation). The simple functors F_{λ} are self-dual under the operator D : F → F^{op}, namely DF_{λ} ≅ F_{λ} as functors, which corresponds to a similar statement for the simple GL_{n}-modules.

The indecomposable injectives in F are therefore indexed by the p-regular partitions. The injective envelope of the functor F_{λ} is written as I_{λ}. Recall that End_{F}(I_{V}) ≅ 𝔽[End V], so that, if dim V = n = λ_{1}, I_{λ} ≅ e_{λ}I_{V}, where e_{λ} is some primitive idempotent in 𝔽[End V]. There is an associated primitive idempotent in 𝔽[GL_{n}], η_{λ}, taken so that 𝔽[GL_{n}]η_{λ} is the projective cover of S_{λ} (which is also an injective envelope of S_{λ}). The embedding 𝔽[GL_{n}] → 𝔽[M_{n}] allows one to regard η_{λ} as an idempotent in 𝔽[M_{n}]; it is no longer primitive but, up to conjugacy, it is expressible as a sum:

η_{λ} = e_{λ} + e\overline{λ},

where \overline{λ} is the p-regular partition λ_{1} − 1 ≥ … ≥ λ_{s} − 1 ≥ 0; this was established in [HK]. This decomposition of η_{λ} corresponds to a direct sum decomposition:

η_{λ}I_{V} ≅ I_{λ} ⊕ I\overline{λ}.

The following functors are also of interest:

Definition 2.3.1.

1. Write Ĩ_{V_{n}} for the functor k_{n−1} I_{V_{n}} ∈ F^{n}. This is the largest sub-functor of I_{V_{n}} which is zero when evaluated on 𝔽^{n−1}.

2. If λ is a p-regular partition with λ_{1} = n, then set Ĩ_{λ} := k_{n−1}I_{λ}. □

It is useful to have another characterization of the functors Ĩ_{V_{n}}, which is given by using the functors R_{n}. Recall that 𝔽[GL_{n}] is regarded as an object in M_{n} via the algebra map 𝔽[M_{n}] → 𝔽[GL_{n}]. The duality functor D : F → F^{op} restricts to a duality functor D : M_{n} → M_{n}^{op}; there is a natural isomorphism in M_{n}:

D𝔽[GL_{n}] ≅ 𝔽[GL_{n}].

To prove this, it is sufficient to look at the GL_{n}-action; D𝔽[GL_{n}] is isomorphic to 𝔽^{GL_{n}} with the transpose action. The latter space has a basis given by the maps α_{g}, for g ∈ GL_{n}, which are defined by x ↦ 0 if x ≠ g and g ↦ 1. GL_{n} acts transitively on this basis (regarded as a set), hence 𝔽^{GL_{n}} is a free permutation module, thus is isomorphic to 𝔽[GL_{n}].

Proposition 2.3.2.

1. Ĩ_{V_{n}} is isomorphic to R_{n}(𝔽[GL_{n}]).

2. End_{F}(Ĩ_{V_{n}}) ≅ 𝔽[GL_{n}] as rings.

3. If λ is a p-regular partition indexing a simple GL_{n}-module S_{λ} and P_{λ} is the projective cover of S_{λ}, then Ĩ_{λ} ≅ R_{n}(P_{λ}).

Proof. The injective I_{V_{n}} is isomorphic to R_{n}(D𝔽[M_{n}]). The map 𝔽[M_{n}] → 𝔽[GL_{n}] in M_{n} dualizes to 𝔽[GL_{n}] ↪ D𝔽[M_{n}], which induces an injection R_{n}(𝔽[GL_{n}]) ↪ I_{V_{n}}; hence it is sufficient to show that R_{n}𝔽[GL_{n}] has the correct universal property.

It is clear that R_{n}𝔽[GL_{n}] ∈ F^{n}, so consider Hom_{F}(G, R_{n}𝔽[GL_{n}]), where G ∈ F^{n}. Using the adjunction (E_{n}, R_{n}), this is isomorphic to Hom_{M_{n}}(G(𝔽^{n}), 𝔽[GL_{n}]), which identifies by duality with Hom_{M_{n}}(𝔽[GL_{n}], DG(𝔽^{n})). Since DG(𝔽^{n}) necessarily belongs to GL_{n} ⊂ M_{n}, this is isomorphic to DG(𝔽^{n}), which establishes the required universal property.

In particular, this argument establishes an isomorphism of vector spaces

End_{F}(Ĩ_{V_{n}}) ≅ 𝔽[GL_{n}].

There is an injection of rings 𝔽[GL_{n}] ↪ End_{F}(Ĩ_{V_{n}}), given by restriction of the action of 𝔽[GL_{n}] ⊂ 𝔽[M_{n}] on I_{V_{n}}. Hence, the above is an isomorphism of rings.

Finally, it is formal that Ĩ_{V_{n}} is injective in F^{n} and splits as a direct sum of functors Ĩ_{λ}. This splitting is achieved by using primitive idempotents η_{λ} ∈ 𝔽[GL_{n}]; then P_{λ} ≅ 𝔽[GL_{n}]η_{λ} and Ĩ_{λ} ≅ η_{λ}Ĩ_{V_{n}}. The stated isomorphism follows by standard arguments.

3. Co-Weyl objects and J-good filtrations

The fundamental objects of study in this paper may now be defined:

Definition 3.0.1. Suppose that λ is a p-regular partition with λ_{1} = n. The associated co-Weyl object J_{λ} is defined to be J_{λ} := R_{n}S_{λ}, where R_{n} is the functor GL_{n} → F^{n} ⊂ F.

It should be observed immediately that J_{λ} is an analytic functor, with simple socle F_{λ} and that J_{λ} embeds in I_{V_{n}}. This may be seen by considering the inclusion S_{λ} ↪ 𝔽[GL_{n}] of GL_{n}-modules, which yields an injection J_{λ} = R_{n}S_{λ} ↪ R_{n}(𝔽[GL_{n}]) ≅ Ĩ_{V_{n}}. As Ĩ_{V_{n}} is a sub-functor of I_{V_{n}}, this shows that J_{λ} embeds in I_{V_{n}}. Moreover, by definition E_{n}J_{λ} ≅ S_{λ}; it follows that J_{λ} has a simple socle F_{λ}, since the factors of the socle of I_{V_{n}} are detected by E_{n}.

The definition of the functor R_{n} then ensures that J_{λ} may be characterized up to isomorphism as being the largest sub-functor F of I_{V_{n}} such that E_{n}F ≅ S_{λ}. (The last property implies that E_{n−1}F = 0.) The following should be noted:

Lemma 3.0.2.

1. If λ is a p-regular partition with λ_{1} = n, then J_{λ} ≅ R_{k}E_{k} J_{λ}, for any k ≥ n.

2. J_{λ} ≅ Hom_{GL_{n}}(S_{λ}, Ĩ_{V_{n}}).

Example 3.0.3. There are two basic examples of co-Weyl functors with which one should be familiar, working over the field 𝔽 = 𝔽_{2}; we consider those induced from simple GL_{n}-modules, for some n.

1. By choice of indexing S_{(n)} ≅ Λ^{n}(𝔽^{n}) ≅ 𝔽 is the trivial GL_{n}-module. In this case J_{(n)} will be denoted by \overline{D}(n); this is due to a relation with the Dickson invariants; namely \overline{D}(n) may be seen to be isomorphic to the functor (Ĩ_{V_{n}})^{GL_{n}}, where GL_{n} acts naturally as endomorphisms of Ĩ_{V_{n}}.

2. Write ⟨n⟩ for the ‘triangular partition’ n > n − 1 > … > 1 > 0 of length n. The module S_{⟨n⟩} is the Steinberg module for GL_{n} and is (up to isomorphism) the only simple module which is projective. The corresponding functor J_{⟨n⟩} is usually written as L(n) and is an injective functor. (The notation derives from a length filtration of certain modules over the Steenrod algebra, which motivated its use in topology [MP]). This is (up to isomorphism) the unique co-Weyl functor induced from a simple GL_{n}-module which is injective.

One should note that, for n = 1, there are isomorphisms L(1) ≅ \overline{I} ≅ \overline{D}(1).

One could also choose to study the ‘Weyl objects’ DJ_{λ} which are the duals of the co-Weyl objects. These are isomorphic to L_{n}S_{λ}, for suitable n. However, these functors are never analytic, for n > 0; one should be aware that a choice has been made here to work with the category of analytic functors.

3.1. Non ˜∇-nilpotence of J_{λ}. An important initial step in the arguments of this paper is given by using the functors ˜∇_{k} to give an approximation to the size of the functors J_{λ}, for varying λ_{1}. It is assumed throughout this section that 𝔽 = 𝔽_{2}.

Recall from [P] that a functor is said to be ˜∇_{k}-nilpotent if there exists some integer N so that ( ˜∇_{k})^{N}F = 0; the full sub-category of ˜∇_{k}-nilpotent functors is thick [P]. Now, ˜∇_{k}I_{V_{n}} = 0 if k > n (by the work of [P]), so that it follows that ˜∇_{k}J_{λ} = 0 for k > λ_{1}, since ˜∇_{k} preserves injections. The key result is:

Proposition 3.1.1. The functor J_{λ}, with λ_{1} = n, is not ˜∇_{n}-nilpotent.

Proof. This relies upon studying the auxiliary functor J_{λ}, which is defined by setting

J_{λ} := Hom_{GL_{n}}(S_{λ}, I_{V_{n}}).

The surjection P_{λ} →→ S_{λ} induces an inclusion J_{λ} ↪ I_{λ}⊕I\overline{λ} ≅ η_{λ}I_{V_{n}} (see Section2).

There are non-trivial GL_{n}-maps 𝔽[GL_{n}] →→ S_{λ} ↪ 𝔽[GL_{n}] which induce maps I_{V_{n}} → J_{λ} → I_{V_{n}}. The composite σ is non-trivial and is induced by an element of the group ring 𝔽[GL_{n}], regarded as a sub-ring of 𝔽[M_{n}] ≅ End(I_{V_{n}}). Theorem 2.1.2 shows that ˜∇_{n}σ = σ, so that ( ˜∇_{n})^{t}σ is non-zero for all t > 0. It follows that ( ˜∇_{n})^{t}J_{λ} is non-zero for all t > 0, so that J_{λ} is not ˜∇_{n}-nilpotent.

There is an exact sequence of functors with GL_{n}-action:

0 → Ĩ_{V_{n}} → I_{V_{n}} → I_{V_{n−1}} ⊗ 𝔽^{hom(𝔽^{n},V_{n−1})},

where the second map is induced by composition

hom(𝔽^{n}, V_{n−1}) × hom(V_{n−1}, −) → hom(𝔽^{n}, −).

Applying the functor Hom_{GL_{n}}(S_{λ}, −) to the exact sequence yields an exact sequence:

0 → J_{λ} → J_{λ} → Q → 0 (1)

where Q is a sub-functor of a finite direct sum of functors I_{V_{n−1}}. Thus, J_{λ} is a sub-functor of J_{λ} and the cokernel J_{λ}/J_{λ} is ˜∇_{n}-nilpotent. (More precisely, the cokernel is zero under ˜∇_{n}.)

The category of ˜∇_{n}-nilpotent functors is shown to be thick in [P]. Thus, the short exact sequence 1, in which Q is ˜∇_{n}-nilpotent and J_{λ} is not ˜∇_{n}-nilpotent shows that J_{λ} is not ˜∇_{n}-nilpotent.

Example 3.1.2. As an immediate application of this result, consider the functor \overline{D}(2) = R_{2}(Λ^{2}(𝔽^{2})) which arises in Remark 2.2.2. The proposition shows that \overline{D}(2) is not ˜∇_{2}-nilpotent; this proves the claimed non-exactness, since the functor \overline{I}/Λ^{1} is a quotient of I_{𝔽} and is therefore ˜∇_{2}-nilpotent, since ˜∇_{2} preserves surjections and ˜∇_{2} I_{𝔽} = 0.

Similarly, one has the following:

Corollary 3.1.3. Suppose that λ is a p-regular partition with λ_{1} = n. The co-Weyl functor J_{λ} is not a sub-quotient of a finite direct sum ⨁_{finite} I_{V_{k}} with k < n.

Proof. The functor ˜∇_{n} preserves injections and surjections; since ˜∇_{n}J_{λ} ≠ 0 and ˜∇_{n}(⨁_{finite} I_{V_{k}}) = 0, the result follows.

3.2. J-good filtrations. The material of this section is analogous to the notions used in the study of highest weight categories, as introduced by Cline, Parhsall and Scott. A brief review of this material is contained in Chapter 4 of [M]. The concept of a J-good filtration corresponds to that of a ∇-good filtration (for a different ∇!) in the setting of highest weight categories. Alternatively, this may be related to the study of good filtrations in algebraic group theory, as discussed in Section 4 of [J, Part II].

Definition 3.2.1. A functor F is J-good if it admits a finite filtration 0 = F_{0} ⊂ F_{1} ⊂ … ⊂ F_{N} = F such that each F_{k}/F_{k−1} is non-zero and isomorphic to some co-Weyl functor J_{λ(k)}. The integer N is termed the J-length of F and such a filtration is said to be J-good.

There is an evident notion of a DJ-good functor; namely G is DJ-good if it takes finite dimensional values and DG is J-good.

A J-good functor F is said to be J^{n}-good if it is isomorphic to R_{n}M, where M is a GL_{n}-module and R_{n} is the functor R_{n} : GL_{n} → F^{n} ⊂ F. The full sub-category of J^{n}-good functors is denoted here by J^{n}. The functor R_{n} : GL_{n} → F^{n} factors through J^{n}, since R_{n}|_{GL_{n}} is exact; in fact one has:

Proposition 3.2.2. The functors E_{n} : J^{n} ⇌ GL_{n} : R_{n} induce an equivalence of categories.

Proof. This is clear at the level of objects. For the morphisms, use the adjunction isomorphism and the isomorphism E_{n}R_{n}M ≅ M to deduce that Hom_{F}(R_{n}M, R_{n}N) ≅ Hom_{GL_{n}}(E_{n}R_{n}M, N) ≅ Hom_{GL_{n}}(M, N).

The defining property of the functors J_{λ} establishes the following result:

Lemma 3.2.3. Suppose that μ, λ are p-regular partitions, with λ_{1} = n. If F is an analytic functor with E_{n}(F) = 0, then Ext_{F}^{1}(F, J_{λ}) = 0. In particular, if μ_{1} > λ_{1}, then Ext_{F}^{1}(J_{μ}, J_{λ}) = 0.

This lemma allows one to re-order any J-good filtration so that the largest co-Weyl objects are at the bottom; recall that k_{n} : F → F^{n+1} is the right adjoint to the inclusion F^{n+1} ↪ F. If F is an analytic functor with a finite socle, there exists t so that k_{t}(F) = 0. In this case, one may form the following natural filtration:

0 = k_{t}(F) ⊂ k_{t−1}F ⊂ … ⊂ k_{0}F ⊂ F. (2)

Combining Lemma 3.2.3 with Proposition 2.2.1, one may conclude the following result:

Proposition 3.2.4. An analytic functor F with finite socle is J-good if and only if each functor k_{s}F/k_{s+1}F in the filtration (2) is J-good. This holds if and only if k_{s}F/k_{s+1}F ≅ R_{s+1}M for some GL_{s+1}-module M.

There is another related result:

Proposition 3.2.5. An analytic functor F with finite socle is J-good if and only if one of the following conditions hold:

1. E_{s−1}F = 0, for some s > 0, the functor k_{s}F is J-good and the sequence

k_{s}F → F → R_{s}E_{s}F

is short exact.

2. For all s, the unit F → R_{s}E_{s}F is surjective.

Proof. The first statement is clear: if E_{s−1}F = 0, then E_{s}F should be regarded as an object in GL_{s} ⊂ M_{s}. Since R_{s}|_{GL_{s}} is an exact functor, it follows that R_{s}E_{s}F is J^{s}-good. By hypothesis, k_{s}F is J-good, so that the result follows.

Consider the second statement; suppose that F is an analytic functor with finite socle and that the unit maps F → R_{s}E_{s}F are surjective, for every s. By the first part of the proposition, it will suffice to show that the maps k_{t}F → R_{t+1}E_{t+1}k_{t}F are surjective for every t, since this will allow a decreasing induction on t.

To prove this, form the commutative diagram:

k_{t}F  →  F  →  R_{t}E_{t}F
↓  ↓  ↓ ι
0 → R_{t+1}E_{t+1}k_{t}F → R_{t+1}E_{t+1}F → R_{t+1}E_{t+1}R_{t}E_{t}F

The top row is a short exact sequence, by the hypothesis and the definition of the functor k_{t}, whereas the bottom row is exact (not necessarily short exact), by the left exactness of R_{t+1}. The vertical arrows are the units of the (E_{t+1}, R_{t+1})-adjunction.

Observe that R_{t+1}E_{t+1}R_{t}E_{t}F is naturally isomorphic to R_{t}E_{t}F, so that the unit map ι is the identity, up to isomorphism. Hence, the bottom row is a short exact sequence. Now, the hypothesis shows that F → R_{t+1}E_{t+1}F is a surjection; since ι is an isomorphism, an easy diagram chase shows that the map k_{t}F → R_{t+1}E_{t+1}k_{t}F is surjective. This completes the proof that F is J-good.

It is left as an exercise to show that, if F is a J-good functor, then all the natural maps F → R_{s}E_{s}F are surjective.

The basic facts about J-good functors are summarized as:

Theorem 3.2.6.

1. Suppose that F → G → H is a short exact sequence of analytic functors; if any two of the functors are J-good, then so is the third.

2. If F ⊕ H is a J-good functor, then so are both F and H.

3. If F, G are J-good sub-functors of an analytic functor H with finite socle, then F ∩ G and F + G are both J-good.

4. If H is an analytic functor with finite socle, then there is a well-defined maximal J-good sub-functor J(H) ↪ H.

Proof. Part 1: It is clear that, if F → G → H is a short exact sequence, with F, H J-good functors, then G is a J-good functor. Hence it remains to consider the case F, G J-good and the case G, H J-good.

Case 1: Suppose that F, G are J-good and choose t minimal so that k_{t}G = 0. By the left exactness of k_{t}, it follows that k_{t}F = 0. Now write Q for the quotient k_{t−1}G/k_{t−1}F, which is J^{t}-good, by Proposition 3.2.4, since the functors k_{t−1}G, k_{t−1}G are both J^{t}-good. The universal property of the cokernel induces a canonical map Q → H; I claim that this map is an injection. The kernel ker{Q → H} injects in F/k_{t−1}F, by standard arguments. By the definition of k_{t−1}F, F/k_{t−1}F embeds in a finite direct sum ⨁_{finite} I_{V_{t−1}}; since Hom_{F}(Q, I_{V_{t−1}}) is zero by construction, this implies that the kernel of Q → H is zero.

Now consider the commutative diagram, in which the rows and columns are short exact sequences:

k_{t−1}F → k_{t−1}G → Q
↓  ↓  ↓
F  →  G  → H
↓  ↓  ↓
F/k_{t−1}F → G/k_{t−1}G → H/Q

Now, H is J-good if H/Q is J-good, since Q is J-good. This allows one to perform an induction upon the J-length of G, since the hypotheses ensure that F/k_{t−1}F and G/k_{t−1}G are both J-good and the J-length of G/k_{t−1}G is less than that of G. The induction is started by the case G = 0.

Case 2: G, H both J-good. This result is proved by an induction upon the J-length of G, the initial case being G = 0, which is trivial. Now choose s so that k_{s}G = 0 and k_{s−1}G ≠ 0. It follows that H embeds in a finite direct sum of I_{V_{s}}’s, since H is J-good and J_{λ}, for λ_{1} > s cannot occur as a sub-quotient of G (which embeds in a finite direct sum ⨁_{finite} I_{V_{s}}), by Proposition 3.2.4.

For the inductive step, there is some J_{μ}, with μ_{1} = s, which embeds in G; by the defining properties of J_{μ} and the choice of s, the composite map J_{μ} → G → H is either injective or trivial.

In the first case, there is a short exact sequence F → G/J_{μ} → H/J_{μ}. In the second, the map J_{μ} → G factors through F and there is a short exact sequence F/J_{μ} → G/J_{μ} → H. In both cases, this forms the basis of the inductive step, since the J-length of G/J_{μ} is less than that of G. (Observe in the second case that F is J-good if and only if F/J_{μ} is J-good.)

Statement 2 may be regarded as an application of Proposition 3.2.5, and is left as an exercise.

To prove the third statement, observe that there is a short exact sequence F ∩ G → F ⊕ G → F + G. This means that we may consider F ∩ G as a sub-functor of F ⊕ G; moreover, it suffices to show that F ∩ G is J-good.

This may be proved by an induction on the J-length of F ⊕G. Choose t maximal so that k_{t−1}F = 0 = k_{t−1}G and suppose (without loss of generality) that k_{t}F ≠ 0. Now, there is some λ with λ_{1} = t so that J_{λ} embeds in F, since F is J-good.

By choice of t, any map F_{λ} ↪ F extends uniquely to a map J_{λ} ↪ F and the same fact holds for G. Suppose that the factor F_{λ} ∈ soc F lies in F ∩ G, then J_{λ} embeds in F ∩ G. Consider the embedding (F ∩ G)/J_{λ} ↪ (F/J_{λ}) ⊕ (G/J_{λ}) and conclude that (F ∩ G)/J_{λ} is J-good, by the induction on the J-length of the direct sum.

Otherwise, there is an embedding (F/J_{λ})∩G ↪ (F/J_{λ})⊕G and the result again follows by an induction on the J-length of the direct sum.

The final statement is an application of the third statement; J(F) is defined as the sum of all the J-good sub-functors of F. (It is not difficult to show that there are only finitely many J-good sub-functors of an analytic functor F with a finite socle.)

4. The indecomposable injectives in F are J-good

The purpose of this section is to prove that the indecomposable injective functors in F are J-good. This has an important application to the Cartan matrices of 𝔽[M_{n}] and 𝔽[GL_{n}].

Theorem 4.0.1. The indecomposable injective functors in F are J-good.

The proof of the theorem given here relies on the following result:

Lemma 4.0.2. The natural map induced by composition,

hom(𝔽^{n}, V) ×_{End(𝔽^{n})} hom(W, 𝔽^{n}) → hom(W, V)

is an injection. The image identifies with hom_{≤n}(W, V), the set of morphisms of rank ≤ n.

Proof. The proof is by induction upon n; the case n = 0 is trivial; to carry out the inductive step, consider the set S_{n} = hom(𝔽^{n}, V) ×_{End(𝔽^{n})} hom(W, 𝔽^{n}) and fix an idempotent e ∈ End(𝔽^{n}), which is a projection onto 𝔽^{n−1}. Suppose that e factors as 𝔽^{n} →→^{p} 𝔽^{n−1} ↪^{i} 𝔽^{n} and observe that there is a natural isomorphism e End(𝔽^{n})e ≅ End(𝔽^{n−1}).

There is an inclusion S_{n−1} = hom(𝔽^{n−1}, V) ×_{End(𝔽^{n−1})} hom(W, 𝔽^{n−1}) ↪ S_{n} which is induced by the maps p : 𝔽^{n} →→ 𝔽^{n−1} and i : 𝔽^{n−1} ↪ 𝔽^{n}. I claim that the complement identifies (as a set) with T := inj(𝔽^{n}, V) ×_{GL_{n}} surj(W, 𝔽^{n}), where inj(−, −) denotes the set of injections and surj(−, −) denotes the set of surjections. Here, T is regarded as a sub-set of S_{n} in the evident way.

To see this, suppose that (α, β) ∈ hom(𝔽^{n}, V) × hom(W, 𝔽^{n}) represents an element in S_{n} which is not in T. Thus, either α is not injective or β is not surjective. Suppose that α is not injective, then there exists γ ∈ hom(𝔽^{n}, V) and r ∈ End(𝔽^{n}) so that α = γer. Hence, (α, β) is equivalent to (γer, β) ∼ (γe, erβ), using the fact that e^{2} = e; thus (α, β) represents an element in S_{n−1}. A similar argument works if β is not surjective.

Hence S_{n} identifies with S_{n−1} ⨿ T; by induction the composition map S_{n−1} → hom(W, V) is injective, whereas the composition map restricted to T may easily be shown to be injective, by the uniqueness of the factorization of a map of rank n through 𝔽^{n}, up to elements of Aut(𝔽^{n}). Conclude that the composition map S_{n} ↪ hom(W, V) is injective. Finally, it is clear that the image of this map is the set hom_{≤n}(W, V) of maps of rank ≤ n.

Proof of Theorem 4.0.1. To prove the result, it suffices to show that the functors I_{V_{n}} are J-good for all n, by Theorem 3.2.6. This is achieved by using the criterion of Proposition 3.2.5. For typographical reasons, it is easiest to dualize and study the projective functors P_{W}, so that the criterion becomes:

• The functor P_{W} is DJ-good if and only if the maps L_{n}E_{n}P_{W} → P_{W} are injective for each n ≥ 0.

One has E_{n}P_{W} ≅ 𝔽[hom(W, 𝔽^{n})], so that

L_{n}E_{n}P_{W} ≅ 𝔽[hom(𝔽^{n}, −)] ⊗_{End(𝔽^{n})} 𝔽[hom(W, 𝔽^{n})];

this is isomorphic to 𝔽[hom(𝔽^{n}, −)×_{End(𝔽^{n})}hom(W, 𝔽^{n})]. The map L_{n}E_{n}P_{W} → P_{W} identifies with the composition map.

Thus, one may appeal to the lemma above, which shows that the composition map hom(𝔽^{n}, V) ×_{End(𝔽^{n})} hom(W, 𝔽^{n}) → hom(W, V) is injective, for all V. Hence, the same is true after ‘linearization’ by 𝔽[−], so that the result is established. The map L_{n}E_{n}P_{W} → P_{W} identifies with the inclusion 𝔽[hom_{≤n}(W, −)] → 𝔽[hom(W, −)].

Remark 4.0.3. Essentially, one is analysing the filtration of P_{W} given by the subfunctors 𝔽[hom_{≤n}(W, −)]; a similar analysis appears in [K4]. It is more direct to prove the theorem by showing that the quotients

𝔽[hom_{≤n}(W, V)]/𝔽[hom_{≤n−1}(V, W)]

are J-good. □

Notation 4.0.4. Write ♯μ(F) for the dimension dim Hom_{F}(F, I_{μ}), which is equal to the number of composition factors of F_{μ} in F. The partitions may be ordered (for example by a lexicographical ordering) so that λ < μ if λ_{1} < μ_{1}. Then one may consider the following matrices of coefficients:

1. ♯μ(I_{λ}): the Cartan matrix for the category F.

2. ♯μ(J_{ν}): the multiplicity matrix for the functor J_{ν}.

3. (α_{λν}): the J-good decomposition matrix, which is defined by the equation:

I_{λ} = α_{λν}J_{ν},

which indicates that I_{λ} has a J-good filtration in which J_{λ} occurs with multiplicity α_{λν}. □

The defining properties of the co-Weyl objects mean that the matrices ♯μ(J_{ν}) and (α_{λν}) have special forms. The matrix ♯μ(J_{ν}) is triangular, with 1’s on the diagonal. Namely:

♯μ(J_{ν}) = \{ \begin{matrix}0 & if μ_{1} ≤ ν_{1} and μ ≠ ν, \\ 1 & if μ = ν. \end{matrix}

The reader is encouraged to write down the block form of this matrix.

Similarly, Proposition 3.2.4 implies that α_{λν} = 0 if ν_{1} > λ_{1}. Moreover, the coefficients α_{λν} for λ_{1} = ν_{1} = n identify with the Cartan matrix C_{λν} for the finite group algebra 𝔽[GL_{n}], by the same result. Again, the reader is encouraged to picture the block form of this matrix.

These matrices are related by the equation:

♯μ(I_{λ}) = α_{λν}♯μ(J_{ν}) (3)

in which a summation over ν is implicit.

This equation shows that the matrix ♯μ(I_{λ}) is determined by the other two matrices; an important fact is that the converse is true:

Proposition 4.0.5. The Cartan invariants ♯μ(I_{λ}) determine the matrices ♯μ(J_{ν}) and (α_{λν}).

Proof. Prove by induction on n that:

Hypothesis stage n: The matrix ♯μ(I_{λ}) determines (α_{λν}) for λ_{1}, ν_{1} ≤ n and ♯μ(J_{ν}) for ν_{1} < n.

The induction starts at n = 0, where the statement is trivial to establish. For the inductive step, suppose that the hypothesis holds for n and start by determining the coefficients ♯μ(J_{ν}) for ν_{1} = n. Equation (3), when λ_{1} = n, may be rewritten as

∑_{ν_{1}=n} C_{λν}♯μ(J_{ν}) = ♯μ(I_{λ}) − ∑_{κ_{1}<n} α_{λκ}♯μ(J_{κ}),

where C_{λν} = α_{λν} for ν_{1} = n identifies with the Cartan matrix for 𝔽[GL_{n}], which is invertible [CR, Theorem 18.25]. The inductive hypothesis implies that the coefficients C_{λν} and all terms on the right hand side are known, hence this equation determines the coefficients ♯μ(J_{ν}).

Now determine the coefficients (α_{λν}) for λ_{1}, ν_{1} ≤ n+1 as follows. The coefficients ♯μ(J_{ν}) are known for μ_{1}, ν_{1} ≤ n + 1, by the special form of this matrix, which has blocks which are identity matrices on the diagonal. Since the matrix is triangular, it is also invertible; hence, the matrix equation

♯μ(I_{λ}) = α_{λν}♯μ(J_{ν})

determines (α_{λμ}) in the given range.

This completes the proof of the inductive step.

It is worth noting that the above result has established the corollary below.

Corollary 4.0.6. The Cartan matrix for 𝔽[M_{n}] determines the Cartan matrix for 𝔽[GL_{n}].

4.1. Example of a J-good filtration. We may consider the Grothendieck group of J-good functors; namely, form the free abelian group (GJ)_{0}F, which is generated by symbols [J_{λ}], where λ ranges over all p-regular partitions. If G is a J-good functor, then write [G] for the element of (GJ)_{0}F which is given inductively as follows: if F → G → H is a short exact sequence of J-good functors, then set [G] = [F] + [H]. The results of Section 5 imply that this may be given a ring structure by the tensor product of functors.

Working in (GJ)_{0}F, when 𝔽 = 𝔽_{2}, the above is illustrated by the study of I_{V_{3}}. Recall that the simple functors F_{λ}, with λ_{1} ≤ 3, are the following:

F_{(0)} ≅ 𝔽,

F_{(1)} ≅ Λ^{1}, F_{(2)} ≅ Λ^{2}, F_{(3)} ≅ Λ^{3} (the exterior power functors),
F_{(2,1)}, F_{(3,2,1)}  (functors induced from Steinberg modules),

F_{(3,2)}, F_{(3,1)}.

The work of [PS] shows that I_{V_{2}} decomposes as

I_{V_{2}} ≅ 𝔽 ⊕ I_{(2)} ⊕ I_{(2,1)}^{⊕2} ⊕ I_{(1)}^{⊕2}.

I_{(2,1)} is isomorphic to the co-Weyl module J_{(2,1)}, which is usually written as L(2), whereas I_{(1)} is usually written as \overline{I}. There is a decomposition of I_{(2)} as [I_{(2)}] = 2[\overline{D}(2)]+[\overline{I}]. To calculate the J-good filtration of I_{V_{3}}, one uses the inductive method of Corollary 4.0.6, starting with the following Cartan matrix for 𝔽[M_{3}] (Table 1), which was taken from [HHS, Theorem 4.2].

Table 1 \begin{aligned}& (0) & (1) & (2, 1) & (2) & (3, 2, 1) & (3, 2) & (3, 1) & (3) \\ (0) & 1 \\ (1) & & 1 & & 1 & & & & 1 \\ (2, 1) & & & 1 & & & 1 & 1 \\ (2) & & 1 & & 3 & & & 2 & 3 \\ (3, 2, 1) & & & & & 1 \\ (3, 2) & & & 1 & & & 4 & 3 & 1 \\ (3, 1) & & & 1 & 2 & & 3 & 6 & 3 \\ (3) & & 1 & & 3 & & 1 & 3 & 5 \end{aligned}

Now, the algorithm may be applied– the details are left to the reader as an exercise; one calculates the results displayed in Table 2, in which the boxed terms correspond to the Cartan matrix for 𝔽[GL_{3}].

4.2. Ext groups for J^{t}-good functors. Using the fact that the injective indecomposable functors in F are J-good, we may now describe Ext groups between J^{t}-good functors.

Table 2 \begin{aligned}& J_{(1)} & J_{(2,1)} & J_{(2)} & J_{(3,2,1)} & J_{(3,2)} & J_{(3,1)} & J_{(3)} \\ I_{(3,2,1)} & & & & 1 \\ I_{(3,2)} & & 1 & & & 3 & 2 & 1 \\ I_{(3,1)} & & 1 & 2 & & 2 & 3 & 1 \\ I_{(3)} & 1 & & 2 & & 1 & 1 & 2 \end{aligned}

Lemma 4.2.1. Suppose that F →→ G is a surjection between J-good functors and that F ↪ R_{t}E_{t}F is an embedding, then the map k_{t−1}F → k_{t−1}G is a surjection.

Proof. The hypothesis ensures that G embeds in R_{t}E_{t}G, since the J-factors of G are J-factors of F and the largest J-factors always occur at the bottom of a J-good filtration. Reduce to the case that k_{t−1}F = 0, by observing that k_{t−1}F, k_{t−1}G are both J^{t}-good functors, so that the image H of the map k_{t−1}F → k_{t−1}G is J-good; thus one can reduce to studying the induced surjection F/k_{t−1}F →→ G/H. Now, F/k_{t−1}F embeds in ⨁_{finite} I_{V_{t−1}}; conclude by Corollary 3.1.3 that G/H does not contain a sub-functor J_{μ}, with μ_{1} = t. It follows immediately that H = k_{t−1}G, establishing the result.

Proposition 4.2.2. Suppose that F, G are J^{t}-good functors, isomorphic respectively to R_{t}M and R_{t}N, for M, N ∈ GL_{t}. Then Ext_{F}^{∗}(F, G) ≅ Ext_{GL_{t}}^{∗}(M, N).

Proof. By Proposition 3.2.2, it is sufficient to show that there is an isomorphism: Ext_{F}^{∗}(F, G) ≅ Ext_{J^{t}}^{∗}(F, G). Consider a minimal injective resolution of G in F, 0 → G → I^{•}. It is not difficult to show that each I^{k} embeds in a finite direct sum ⨁ I_{V_{t}}. Moreover, since k_{t−1} is a right adjoint, k_{t−1}I^{•} is a complex of injectives in F^{t}. Since J^{t} is a full sub-category of F^{t}, it suffices to show that the complex 0 → G → k_{t−1}I^{•} is exact.

This is established by using the fact that each functor I^{k} is J-good. The injective resolution breaks into a Yoneda product of short exact sequences Z_{n} → I^{n} → Z_{n+1}, in which the functors Z_{n} are J-good, since the category of J-good functors is thick. The lemma applies to show that the sequences k_{t−1}Z_{n} → k_{t−1}I^{n} → k_{t−1}Z_{n+1} are all short exact.

5. A homological characterization of J-good functors

The purpose of this section is to give a homological characterization of a J-good functor. The main result is:

Theorem 5.0.1. An analytic functor F with finite socle is J-good if and only if it admits an injective resolution 0 → F → I^{•} such that there exists an integer d so that each I^{k} embeds in a finite direct sum ⨁_{finite} I_{V_{d}}.

Suppose that F is an analytic functor with a finite socle; there is a minimal N so that F embeds in R_{N}E_{N}F (equivalently, k_{N} F = 0). Write this N as emb F, standing for embedding dimension of F. It is clear that emb F ≤ N if and only if F embeds in a finite direct sum ⨁_{finite} I_{V_{N}}.

By Theorem 3.2.6, there is a well-defined notion of a maximal J-good functor J(F) of an analytic functor F with finite socle. This allows one to take the following definition:

Definition 5.0.2. If F is not a J-good functor, then write d_{J}(F) := emb F/J(F).

The proof of the main result of this section uses the interplay between d_{J}(F) and emb F in injective resolutions of functors. The crucial step is the following:

Lemma 5.0.3. Suppose that F is an analytic functor with finite socle and I_{F} is the injective envelope of F. If F is not J-good, then d_{J}(I_{F} /F) > d_{J}(F).

Proof. Consider the embedding F ↪ I_{F}; by hypothesis, the functor F is not Jgood, so that d_{J}(F) = n, for some n > 0. Now, using the fact that I_{F} is J-good, we may consider the J-good functor G = k_{n}(I_{F}) = ker {I_{F} → R_{n}E_{n}I_{F} }. Take F̂ to be the functor F̂ := F +G. It is clear that J(F̂) = G+J(F), so that it follows that d_{J}(F̂) = d_{J}(F). Moreover, a similar reasoning shows that d_{J}(I_{F} /F̂) = d_{J}(I_{F} /F), using the choice of n.

Hence, it suffices to prove the result for F̂. By construction, I_{F} /G embeds in a finite direct sum ⨁_{finite} I_{V_{n}}. Moreover, by the choice of n, there is an embedding J_{λ} ↪ I_{F} /G, for some λ with λ_{1} = n, so that the composite J_{λ} → I_{F} /G → I_{F} /F̂ is non-trivial.

This implies, by the defining property of J_{λ}, that soc(I_{F} /F̂) contains a composition factor F_{ν}, with ν_{1} > n, which lies in the image of the above map. Thus, to prove the result, it suffices to show that J_{ν} does not embed in I_{F} /F̂. This follows by applying Corollary 3.1.3, since I_{F} /F̂ is a quotient of I_{F} /G, which embeds in ⨁_{finite} I_{V_{n}}, so that I_{F} /F̂ is a sub-quotient of ⨁_{finite} I_{V_{n}}.

Proof of Theorem 5.0.1. If F is J-good, then F embeds in a finite direct sum I^{0} = ⨁_{finite} I_{W}, where dim(W) = emb F. The functor I^{0}/F is J-good, by Theorem 3.2.6, with finite socle and emb(I^{0}/F) ≤ emb F. Thus, an inductive argument shows that, if F is J-good, then F has an injective resolution of the form given.

For the converse, suppose that F is an analytic functor which admits an injective resolution of the form given; thus, we may assume that F has a minimal injective resolution:

0 → F → I^{0} → I^{1} → … → I^{k} → …,

where each I^{k} has finite socle and emb I^{k} ≤ d, for some fixed d.

Write Z_{n+1} for the image of I^{n} → I^{n+1}, so that I^{n+1} is the injective envelope of Z_{n+1} ≅ I^{n}/Z_{n}. In particular, emb I^{n+1} = emb(I^{n}/Z_{n}).

Now, suppose that F is not J-good; then, since there are short exact sequences Z_{n} → I^{n} → Z_{n+1}, Theorem 3.2.6 shows that each Z_{n} is not J-good, since I^{n} is J-good by Theorem 4.0.1. This means that Lemma 5.0.3 may be applied, showing that d_{J}(I^{n}/Z_{n}) > d_{J}(Z_{n}). In particular, since I^{n}/Z_{n} ≅ Z_{n+1}, this shows that d_{J}(Z_{n}) is not bounded as n increases.

Now, if F is an analytic functor with finite socle, Proposition 3.2.4 implies that emb F ≥ d_{J}(F). Hence, the above argument shows that emb(Z_{n}) is not bounded. Since I^{n} is the injective envelope of Z_{n}, it follows that emb I^{n} is not bounded, which is a contradiction.

The form of the injective resolution of a J-good functor F is determined by an injective resolution of E_{d}F as an M_{d}-module, if d is sufficiently large:

Proposition 5.0.4. Suppose that F is a J-good functor with emb F = d. If n ≥ d and J^{•} is an injective resolution of E_{n}F in M_{n}, then R_{n}J^{•} is an injective resolution of F.

An important corollary to the theorem is that this allows one to show that certain constructions between J-good functors give J-good functors:

Corollary 5.0.5.

1. If F, G are J-good functors then F ⊗ G is J-good.

2. If F is a J-good functor then ΔF is J-good.

Proof. The first statement follows by tensoring injective resolutions of F, G which are of the form supplied by Theorem 5.0.1. Since I_{V_{m}} ⊗ I_{V_{n}} is isomorphic to I_{V_{m+n}}, the result then follows by a second application of Theorem 5.0.1.

For the second statement, one applies Δ to an injective resolution of F, as supplied by the theorem. Since Δ is an exact functor and ΔI_{V_{n}} is a finite direct sum of I_{V_{n}}’s, the result follows, as before.

Recall that (GJ)_{0}F is taken to denote the Grothendieck group of J-good functors. This abelian group has a product which is induced by the tensor product; namely [F].[G] = [F ⊗ G], which gives (GJ)_{0}F a ring structure. The additive identity is [0], whereas the multiplicative identity is [𝔽].

Remark 5.0.6. The author intends to study the structure of this ring in a future paper; at present this is not determined.

5.1. Example. The example below continues the study of the structure of I_{V_{3}}, when 𝔽 = 𝔽_{2}, started in Section 4. The reader is referred to this section for the list of simple functors F_{λ} with λ_{1} ≤ 3. The above results show that the functor \overline{D}(2)⊗\overline{I} is J-good and it is a sub-functor of I_{V_{3}}. It may be studied in the Grothendieck ring (GJ)_{0}F; one must find the multiplicity of each co-Weyl object.

Notation 5.1.1. If F ∈ F, then write [F] ∼ ∑ a_{λ}[F_{λ}], for positive integers a_{λ}, if there is equality in the Grothendieck ring G_{0}F, modulo factors F_{μ} with μ_{1} ≥ 4.

To perform the calculation it is necessary to calculate (up to the relation ∼) the functors F_{λ} ⊗F_{μ} for the simple functors appearing above. This may be done, giving the following:

[Λ^{1} ⊗ Λ^{2}] ∼ [Λ^{3}] + [F_{(2,1)}]
[Λ^{2} ⊗ Λ^{2}] ∼ 2[F_{(3,1)}] + [Λ^{2}]
[Λ^{3} ⊗ Λ^{2}] = [Λ^{2} ⊗ Λ^{3}] ∼ [F_{(3,2)}]
[Λ^{1} ⊗ Λ^{3}] ∼ [F_{(3,1)}]
[Λ^{3} ⊗ Λ^{3}] ∼ [Λ^{3}]
[Λ^{1} ⊗ F_{(3,1)}] ∼ [F_{(3,1)}] + 2[F_{(3,2)}]
[Λ^{2} ⊗ F_{(3,1)}] ∼ [Λ^{3}] + [F_{(3,2,1)}]
[Λ^{3} ⊗ F_{(3,1)}] ∼ [F_{(3,1)}]

For this, a number of techniques are used. For the tensor product of two exterior powers, use of the functor ∇_{2} is invaluable. For the remaining cases, one may study the GL_{3}-module obtained by evaluating on 𝔽^{3}; the given decompositions then become clear. One concludes, by ‘subtracting’ the composition factors which occur in J_{(2)} and J_{(2,1)}, that:

[\overline{D}(2) ⊗ \overline{I}] = [J_{(2)}] + [J_{(2,1)}] + 2[J_{(3)}] + 3[J_{(3,1)}] + 3[J_{(3,2)}] + [J_{(3,2,1)}].

6. Proof of the simplicity theorem

This section is devoted to the proof of the theorem below; throughout the section, 𝔽 is taken to be 𝔽_{2}, the prime field with two elements. Recall that an analytic functor F is said to be ˜∇_{n}-nilpotent if there is some k so that ( ˜∇_{n})^{k}F = 0.

Theorem 6.0.1. Suppose that 𝔽 = 𝔽_{2} and that λ is a 2-regular partition with λ_{1} = n. If F is a proper sub-functor of J_{λ} then F is ˜∇_{n}-nilpotent.

The result is established by proving the stronger result:

Proposition 6.0.2. Suppose that J_{λ} is as above. If f : J_{λ} → G is a non-trivial map, then there exists an integer k so that ( ˜∇_{n})^{k}f is injective.

Proof of Theorem 6.0.1 (Assuming Proposition 6.0.2). If F is a proper sub-functor of J_{λ} then one may form the quotient map J_{λ} → J_{λ}/F, which is non-trivial, by the hypothesis. Thus, Proposition 6.0.2 shows that there exists some k so that ( ˜∇_{n})^{k}f is injective. Consider the sequence of maps:

( ˜∇_{n})^{k}F → ( ˜∇_{n})^{k}J_{λ} → ( ˜∇_{n})^{k}(J_{λ}/F).

The first map is injective, since ˜∇_{n} preserves injections, and the second map is injective by the choice of k. However, the composite is zero, so that it follows that ( ˜∇_{n})^{k}F = 0, as required.

The proof of Proposition 6.0.2 is facilitated by calculating ˜∇_{n}J_{λ}. This may be done, using the understanding of J-good functors which has already been established.

Lemma 6.0.3. If λ is a 2-regular partition with λ_{1} = n and λ ≠ ⟨n⟩, then ˜∇_{n}J_{λ}(𝔽^{n−1}) = 0.

Proof. Recall that J_{λ}(𝔽^{n}) ≅ F_{λ}(𝔽^{n}) and consider the short exact sequence F_{λ} → J_{λ} → J_{λ}/F_{λ}. Applying ˜∇_{n} yields a short exact sequence

˜∇_{n}F_{λ} → ˜∇_{n}J_{λ} → C,

in which C is a quotient of Δ(J_{λ}/F_{λ}). Since (J_{λ}/F_{λ})(𝔽^{n}) = 0, it follows that Δ(J_{λ}/F_{λ})(𝔽^{n−1}) = 0, which gives C(𝔽^{n−1}) = 0. Hence

˜∇_{n}J_{λ}(𝔽^{n−1}) ≅ ˜∇_{n}F_{λ}(𝔽^{n−1});

the latter is zero, by the results of [P] on the action of ˜∇_{n} on simple functors, unless λ = ⟨n⟩, in which case it is F\overline{λ}(𝔽^{n−1}).

Proposition 6.0.4. Suppose that λ is a 2-regular partition with λ_{1} = n. If λ = ⟨n⟩, then J_{⟨n⟩} ≅ L(n) and ˜∇_{n}L(n) = L(n) ⊕ L(n − 1). Otherwise ˜∇_{n}J_{λ} = J_{λ}.

The proof uses the following lemma, which follows from Corollary 3.1.3:

Lemma 6.0.5. Suppose that J_{λ}, for λ_{1} = n, is a sub-quotient of a sub-functor F of I_{V_{n}}; then this sub-quotient J_{λ} occurs as a sub-quotient of k_{n−1}F ⊂ Ĩ_{V_{n}}

Proof of Proposition 6.0.4. One starts by establishing that J_{λ} is a sub-functor of ˜∇_{n}J_{λ}. Recall that, by construction, ˜∇_{n}J_{λ} is a quotient of ΔJ_{λ}, which is J-good, by Corollary 5.0.5. Also, ΔJ_{λ}(𝔽^{n−2}) = 0, so that ΔJ_{λ} only has J-factors of the form J_{ν}, with ν_{1} ∈ {n, n − 1}. Hence, ΔJ_{λ} occurs as an extension

K → ΔJ_{λ} → H,

in which K is J^{n}-good and H is J^{n−1}-good. Since ˜∇_{n}J_{λ} is not ˜∇^{n}-nilpotent, it follows from Corollary 3.1.3 that the map K → ˜∇_{n}J_{λ} is non-trivial, since otherwise ˜∇_{n}J_{λ} would be the quotient of a ˜∇_{n}-nilpotent functor. Now ˜∇_{n}J_{λ} embeds in J_{λ} ⊂ I_{λ} ⊕ I\overline{λ}, so that the basic properties of the co-Weyl objects show that J_{λ} is a sub-functor of ∇_{n}J_{λ}. It remains to show that the inclusion J_{λ} ↪ ˜∇_{n}J_{λ} is an isomorphism (in the case λ ≠ ⟨n⟩).

Consider the short exact sequence J_{λ} → I_{λ} → Q_{λ}, in which all the functors are J-good. Applying ˜∇_{n} gives a sequence (not necessarily exact in the middle):

˜∇_{n}J_{λ} ↪ I_{λ} ⊕ I\overline{λ} →→ ˜∇_{n}Q_{λ}.

Since J_{λ} ⊂ ˜∇_{n}J_{λ}, the second map factors through (I_{λ}/J_{λ}) ⊕ I\overline{λ}.

Now, Q_{λ} is a sub-functor of ⨁_{finite} I_{V_{n−1}}, so that ˜∇_{n}Q_{λ} is a sub-functor of ⨁_{finite} I_{V_{n−1}}; hence, Lemma 6.0.5 may be applied. If J_{μ} (for μ_{1} = n) is a J-factor of Q_{λ} with multiplicity β_{μ}, then it follows that J_{μ} occurs as a sub-quotient of k_{n−1}( ˜∇_{n}Q_{λ}) with multiplicity β_{μ}.

I claim that the induced map Ĩ_{λ}/J_{λ} → ˜∇_{n}Q_{λ} is injective; write G for the image of this map, which is J-good, since the image of any map from a J^{n}-good functor to ⨁_{finite} I_{V_{n}} is J-good.

To prove injectivity, pass to the quotient map (I_{λ}/Ĩ_{λ}) → ( ˜∇_{n}Q_{λ})/G. The domain is a sub-functor of ⨁_{finite} I_{V_{n−1}}, hence does not contain a sub-quotient J_{μ}, when μ_{1} = n. Thus, G must contain J_{μ} with multiplicity β_{μ}. Now, by definition, β_{μ} is equal to the multiplicity of J_{μ} in Ĩ_{λ}/J_{λ}, so this establishes the claim.

Finally, since ˜∇_{n}J_{λ} embeds in Ĩ_{λ}, by Lemma 6.0.3, and the map Ĩ_{λ}/J_{λ} → ˜∇_{n}Q_{λ} is injective, it follows that J_{λ} = ˜∇_{n}J_{λ}, as required.

Proof of Proposition 6.0.2. The case λ ≠ ⟨n⟩ is treated; the remaining case is obtained by a straightforward adaptation of the argument.

Reduce to considering maps g : J_{λ} → I_{W} with dim(W) ≥ n as follows; if f : J_{λ} → G is a non-trivial map then there exists a map G → I_{W} so that the composite J_{λ} → G → I_{W} is non-trivial, since the image of f is an analytic functor. Moreover, one can take dim(W) ≥ n, by the defining properties of J_{λ}.

One may fix a standard inclusion ι_{λ} of J_{λ} in I_{V_{n}} as follows; namely, take ι_{λ} to be the composite J_{λ} ↪ Ĩ_{V_{n}} → I_{V_{n}}, where the first map is induced from a surjection 𝔽[GL_{n}] → S_{λ} of GL_{n}-modules by applying the functor Hom_{GL_{n}}(−, Ĩ_{V_{n}}). Proposition 6.0.4 shows that ˜∇_{n}J_{λ} ≅ J_{λ} (since λ ≠ ⟨n⟩); an application of Theorem 2.1.2 shows that the map ˜∇_{n}(ι_{λ}) may be identified with ι_{λ}.

Form an extension of g along the map J_{λ} ↪^{ι_{λ}} I_{V_{n}}, using the injectivity of I_{W}. This gives a commutative diagram:

\xymatrix{J_{λ} \ar[dr]_{g} \ar[r]^{ι_{λ}} & I_{V_{n}} \ar[d]^{ϕ} \\  & I_{W} .}

Any map ϕ ∈ Hom_{F}(I_{V_{n}}, I_{W}) may be decomposed uniquely as follows, using the isomorphism Hom_{F}(I_{V_{n}}, I_{W}) ≅ 𝔽[hom(V_{n}, W)]. Namely, take the decomposition hom(V_{n}, W) ≅ hom_{<n}(V_{n}, W) ⨿ rank_{n}(V_{n}, W). The set rank_{n}(V_{n}, W) decomposes as the union of free right GL_{n}-sets, with one orbit for each n-plane in W; the set of n-planes in W is written as Gr_{n}(W), corresponding to the Grassmannian. Then, if ϕ ∈ Hom_{F}(I_{V_{n}}, I_{W}), one may write

ϕ = ϕ_{0} + ∑_{π∈Gr_{n}(W)} ϕ_{π},

where ϕ_{0} ∈ 𝔽[hom_{<n}(V_{n}, W)] and each ϕ_{π} is the sum of rank n maps which are in the conjugacy class of π. The importance of this approach is shown by the following facts, which follow from the defining properties of J_{λ}:

1. ϕ_{0} is zero when restricted to J_{λ}.

2. A map ϕ_{π} : I_{V_{n}} → I_{W} is either injective or zero when restricted to J_{λ}. The second point follows by observing that a map ϕ_{π} may be factored as I_{V_{n}} → I_{V_{n}} → I_{W}, in which the second map is an injection.

Define the λ-length of a map ϕ : I_{V_{n}} → I_{W} (with respect to ι_{λ}) to be the number of π ∈ Gr_{n}(W) such that J_{λ} ↪^{ι_{λ}} I_{V_{n}} →^{ϕ_{π}} I_{W} is non-zero (hence injective).

The proof of the proposition is by induction on the λ-length of ϕ. If ϕ has λlength 1, where g = ϕ|_{J_{λ}}, then g is necessarily injective, so that it suffices to take k = 0.

For the inductive step, suppose that the λ-length of ϕ is greater than one and consider the map ∇_{n}ϕ : I_{V_{n}} ≅ ∇_{n}I_{V_{n}} → Λ^{n}(W) ⊗ I_{W} ≅ ∇_{n}(I_{W}). This map identifies with

∑_{π∈Gr_{n}(W)} v_{π} ⊗ ϕ_{π},

where v_{π} ∈ Λ^{n}(W) is the image under Λ^{n}(π) of the non-zero vector in Λ^{n}(𝔽^{n}).

Since the λ-length of ϕ is greater than one, there are σ, τ ∈ Gr_{n}(W) so that ϕ_{σ}ι_{λ} and ϕ_{τ} ι_{λ} are both injective. Since σ, τ are distinct planes in W, the vectors v_{σ} and v_{τ} are distinct; more particularly, there is a surjection W →→ 𝔽^{n} so that Λ^{n}(β)v_{σ} ≠ 0 and Λ^{n}(β)v_{τ} = 0.

Now, consider the map Λ^{n}(β) ⊗ I_{W} : Λ^{n}(W) ⊗ I_{W} → I_{W}, which fits into the commutative diagram:

\xymatrix{J_{λ} \ar[dr]_{∇_{n}g} \ar[r]^{ι_{λ}} & I_{V_{n}} \ar[d]^{∇_{n}ϕ} \ar[dr]^{ψ} \\  & Λ^{n}(W) ⊗ I_{W} \ar[r] & I_{W} .}

By construction, ψ = ∑_{π∈Gr_{n}(W)|Λ^{n}(β)v_{π}=1} ϕ_{π}; by the choice of β it follows that the λ-length of ψ satisfies 1 ≤ λ − length ψ < λ − length ϕ. Hence, by induction, there is a k so that (∇_{n})^{k}ψ is injective. Applying (∇_{n})^{k} to the above diagram shows that (∇_{n})^{k+1}g is injective, which is the required result.

Recall that there is a unique non-trivial map J_{λ} ↪ I_{λ}.

Corollary 6.0.6. Suppose that I_{λ} →^{f} G is a map so that the composite g : J_{λ} ↪ I_{λ} → G is non-trivial. There exists a k so that (∇_{n})^{k}f is injective.

Proof. We may assume that λ ≠ ⟨n⟩, since I_{λ} = L(n) = J_{⟨n⟩} in this case. Proposition 6.0.2 implies that there is a k so that (∇_{n})^{k−1}g is injective; now (∇_{n})^{k−1}g factors through (∇_{n})^{k−1}I_{λ} ≅ I_{λ} ⊕ I\overline{λ} and (∇_{n})^{k−1} of the inclusion J_{λ} → I_{λ} must identify with the unique inclusion J_{λ} ↪ I_{λ} ⊕ I\overline{λ}. It follows that (∇_{n})^{k−1}f is injective when restricted to I_{λ}.

To complete the proof, observe that ∇_{n} of the inclusion I_{λ} ↪ I_{λ} ⊕ I\overline{λ} is the identity on I_{λ} ⊕ I\overline{λ}. It follows immediately that (∇_{n})^{k}f is injective.

6.1. Consequences of the simplicity theorem. The full sub-category ˜∇_{n}−N il ⊂ F_{ω} of functors which are nilpotent under ˜∇_{n} is shown to be thick in [P]; thus, one may study the category localized away from the ˜∇_{n}-nilpotent functors. For the purposes of this paper, write C_{n} for the category ˜∇_{n}−N il, regarded as a full sub-category of F_{ω}.

Recall the construction of the quotient category F_{ω}/C_{n}, as described in [G]. The objects in the category are the same as those of F_{ω}: namely the analytic functors, whereas, for X, Y ∈ Ob F_{ω}:

Hom_{F_{ω}/C_{n}}(X, Y) ≅ lim_{X′,Y′→} Hom_{F}(X′, Y/Y′),

where the limit is taken over those X′ ⊂ X and Y′ ⊂ Y such that X/X′ and Y′ are objects in C_{n} (ie they are ˜∇_{n}-nilpotent). The work of [G, Chapter 3] shows that F_{ω}/C_{n} is an abelian category; moreover, an object S in the quotient category is simple if and only if whenever A ↪ S is an inclusion in F_{ω}, either A is in C_{n} or the cokernel of A ↪ S is in C_{n}.

Proposition 6.1.1.

1. The functors J_{λ}, where λ_{1} = n, are simple objects in the quotient category F_{ω}/C_{n}.

2. If λ_{1} = μ_{1} = n, then J_{λ} and J_{μ} are isomorphic in F_{ω}/C_{n} if and only if λ = μ.

3. The endomorphism ring of J_{λ}, End_{F_{ω}/C_{n}}(J_{λ}) is 𝔽.

Proof. The first statement is a straightforward corollary to Theorem 6.0.1. To prove the second, consider a non-zero map in F_{ω}/C_{n} between J_{λ} and J_{μ}. This is

represented by a map F →^{f} J_{μ}/G, where F ⊂ J_{λ}, G ⊂ J_{μ} and where both J_{λ}/F and G are ˜∇_{n}-nilpotent.

Theorem 6.0.1 implies immediately that F = J_{λ}, using the fact that J_{λ} is not ˜∇-nilpotent and the category C_{n} is thick. Now apply Proposition 6.0.2 to consider the maps f : J_{λ} → J_{μ}/G and the natural surjection J_{μ} → J_{μ}/G; conclude that there exists k ≥ 0 so that both ( ˜∇_{n})^{k}f : ( ˜∇_{n})^{k}J_{λ} → ( ˜∇_{n})^{k}(J_{μ}/G) is injective and ( ˜∇_{n})^{k}(J_{μ}/G) is isomorphic to ( ˜∇_{n})^{k}J_{μ}. Since ( ˜∇_{n})^{k}J_{μ} always embeds in I_{μ} ⊕ I\overline{μ}, it follows that ( ˜∇_{n})^{k}f induces an injection ( ˜∇_{n})^{k}J_{λ} ↪ I_{μ} ⊕ I\overline{μ}; J_{λ} is a sub-functor of ( ˜∇_{n})^{k}J_{λ}, so that it follows that λ = μ.

Finally, in the case that λ = μ, if p : J_{λ} →→ J_{λ}/G denotes the natural surjection, then an adaptation of the above argument shows that ( ˜∇_{n})^{k}f identifies with ( ˜∇_{n})^{k}p for k sufficiently large. Thus, ( ˜∇_{n})^{k}(f − p) is zero, which implies that f = p, by Proposition 6.0.2. The natural surjections J_{λ} → J_{λ}/G, as G varies amongst proper sub-functors of J_{λ} all represent the same homomorphism in the localized category. This completes the proof of the result.

Remark 6.1.2. These simple objects in F_{ω}/C_{n} are far from being the only simple objects in the quotient category. For example, the results of [P1] construct a nonfinite family of simple objects in the category F_{ω}/C_{1}, which occur as sub-functors of \overline{I} ⊗ Λ^{t} as t varies.

7. Boolean algebras and module structures

The purpose of this section is to show that many J-good functors arise naturally; let B denote the category of 𝔽-Boolean algebras. B is equivalent to the opposite of the category PS of profinite sets: if B is an 𝔽-Boolean algebra, then write spec B for the set of algebra maps Alg(B, 𝔽). This is a profinite set, since B is the colimit of its finitely-generated sub-algebras. There is a natural isomorphism 𝔽^{Spec B}, where this is the set of continuous set maps from Spec B to 𝔽.

Write B^{E_{f}} for the category of functors E_{f} → B; this is regarded as a sub-category of F, namely the category of Boolean algebra valued functors. This category is of fundamental interest due to the connection with the study of unstable algebras over the Steenrod algebra, as established by [HLS, Part II].

There is an equivalence of categories between B^{E_{f}} and Func{E_{f}^{op} → PS}. Suppose that γ : E_{f}^{op} → PS is such a functor; the associated functor in F is given by V ↦ 𝔽^{γ(V)}.

There is an elegant translation of the notion of transcendence degree for an algebra to the category B^{E_{f}}, which is established in [HLS]. For the purposes of this paper, the following restricted notion is sufficient.

Definition 7.0.1. A functor K ∈ B^{E_{f}} which takes finite-dimensional values has transcendence degree ≤ d if K ≅ R_{d}(𝔽^{S}), for some finite right End(V_{d})-set S. Equivalently, K is represented by the functor E_{f}^{op} → PS: V ↦ S ×_{End(V_{d})} hom(V, V_{d}). □

Example 7.0.2. Examples of Boolean algebra functors of transcendence degree ≤ d have already appeared in this paper; the functors I_{𝔽^{d}}, Ĩ_{𝔽^{d}} ⊕ 𝔽 and \overline{D}(d) ⊕ 𝔽 may all be regarded as such functors.

I_{𝔽^{d}} is represented by the right End(V_{d})-set hom(𝔽^{d}, V_{d}^{∗}), where the End(V_{d})- action is the transpose of the usual action on hom(𝔽^{d}, V_{d}). 𝔽 ⊕ Ĩ_{𝔽^{d}} is represented by the End(V_{d})-set which is obtained by collapsing all the singular endomorphisms to a point in the above set. Hence, this identifies with ∗ ⨿ GL_{d}, with an End(V_{d})- action which is induced by the above quotient. Finally, 𝔽 ⊕ \overline{D}(d) is represented by the smallest possible non-trivial End(V_{d})-set, namely X_{d} = ∗_{0} ⨿ ∗_{1}, where ∗_{0} is a sub End(V_{d})-set.

The functor I_{𝔽^{d}} is injective in the category B^{E_{f}} and is determined by the representing property that follows: if γ is a functor E_{f}^{op} which represents K ∈ B^{E_{f}}, then Hom_{B^{E_{f}}} (K, I_{𝔽^{d}}) ≅ γ(𝔽^{d} ^{∗}).

This allows the statement of the main result of this section:

Theorem 7.0.3. Suppose that K ∈ B^{E_{f}} takes finite-dimensional values and has finite transcendence degree, then K is J-good as a functor in F.

Proof. The proof is by induction upon d, the transcendence degree of K. The case d = 0 is easy, since the functor is then a constant functor (with Boolean algebra structure) and is therefore J-good. The first step in the proof is to observe that we may reduce to the case K(0) = 𝔽 and perform the induction on such functors.

Suppose that the result holds for L ∈ B^{E_{f}}, with L(0) = 𝔽 and the transcendence degree of L is less than d and consider K with transcendence degree exactly d. Hence, there is some minimal k ≤ d so that K(𝔽^{k}) ≠ K(0).

In the case that k = d and the End(V_{d})-set representing K is S, then 𝔽^{S} ≅ 𝔽⊕\overline{𝔽^{S}} as an End(V_{d})-module, where every singular endomorphism acts trivially on \overline{𝔽^{S}}. Hence R_{d}(𝔽^{S}) ≅ 𝔽 ⊕ R_{d}(\overline{𝔽^{S}}) is J-good, by Proposition 2.2.1.

This allows an induction on the number of elements in S: as hypothesis, suppose that the theorem is proved for all K induced from End(V_{d})-sets with strictly fewer than N elements. The initial stage of the induction is the trivial case when N = 1.

Now suppose that |S| = N and consider the number k defined above; the case k = d is covered by the previous discussion. If k < d, then there is a non-trivial map in B^{E_{f}}, α : K → I_{𝔽^{k}}, which allows one to perform the inductive step on N.

The kernel L of α is a functorial Boolean algebra ideal, so that 𝔽⊕L is naturally an object in B_{f}^{E}. Since the map α is of the form R_{d}(𝔽^{S}) → R_{d}(hom(𝔽^{k}, V_{d}^{∗})), it follows that 𝔽 ⊕ L ≅ R_{d}(𝔽^{T}), for some End(V_{d})-set T with |T | < |S|, using the left exactness of R_{d}. Hence, by the inductive hypothesis, 𝔽 ⊕ L (and hence L) is J-good.

The image A of the map α : K → I_{𝔽^{k}} is a Boolean algebra valued functor; it is a standard argument (essentially by Yoneda’s lemma) to show that A ⊂ I_{𝔽^{k}} has transcendence degree ≤ k, Thus, by induction, A is J-good, using the fact that k < d. It follows that K is J-good, completing the induction upon N and thus the induction upon d.

This theorem provides us with evidence that J-good functors occur naturally in the study of the category F, since the Boolean algebra valued functors have an importance stressed by [HLS].

The following result may be compared with the results of Henn, [H], who works entirely within the category of unstable modules over the Steenrod algebra, in a more general situation.

Corollary 7.0.4. Every object K ∈ B^{E_{f}} which takes finite-dimensional values and which has finite transcendence degree ≤ d has an injective resolution 0 → K → I^{•}, in which each I^{•} is a direct summand of a finite direct sum ⨁ I_{V_{d}}.

Proof. This follows directly by combining Theorem 7.0.3 with the easier implication of Theorem 5.0.1.

Example 7.0.5. Consider the Dickson algebra functor D(n) := (I_{𝔽^{n}})^{GL_{n}}. This is a Boolean algebra valued functor which may be seen to be represented by the End(V_{n})-set hom(𝔽^{n}, V_{n}^{∗})/GL_{n}. This set is filtered by the rank of the maps (as an End(V_{n})-set). The set of points represented by a map of rank k is

rank_{k}(𝔽^{n}, V_{n}^{∗})/GL_{n},

which identifies with Gr_{k}(V_{n}^{∗}), the set of k-planes in V_{n}^{∗}. This corresponds to the fact that the functor D(n)/𝔽 admits a unique J-good filtration, in which \overline{D}(k) occurs once, for 1 ≤ k ≤ n. The factor 𝔽 ≅ \overline{D}(0) occurs as a direct summand of the functor D(n).

7.1. Module structures for the J_{λ}. Throughout this section, it is supposed that 𝔽 = 𝔽_{2}, the prime field with two elements. In this case, Λ^{n}(𝔽^{n}) is isomorphic to the trivial GL_{n}-module 𝔽; thus, for any simple GL_{n}-module S_{λ}, there is an isomorphism in GL_{n}, Λ^{n}(𝔽^{n}) ⊗ S_{λ} ≅ S_{λ}. Recall that the co-Weyl object J_{λ} is defined as R_{n}(J_{λ}); it follows that there is a unique non-trivial map R_{n}(Λ^{n}(𝔽^{n})) ⊗ R_{n}(S_{λ}) → R_{n}(S_{λ}), by using the (E_{n}, R_{n})-adjunction and the fact that E_{n} commutes with the tensor product. This argument extends to show that this unique non-trivial map, which identifies as a map μ_{λ} : \overline{D}(n) ⊗ J_{λ} → J_{λ} fits into a commutative diagram:

\overline{D}(n) ⊗ \overline{D}(n) ⊗ J_{λ} ⟶^{\overline{D}(n)⊗μ_{λ}}  \overline{D}(n) ⊗ J_{λ}
μ⊗J_{λ} ↓  ↓ μ_{λ}
\overline{D}(n) ⊗ J_{λ}  ⟶^{μ_{λ}}  J_{λ};

this shows that J_{λ} is a \overline{D}(n)-module, in an obvious sense, since the product \overline{D}(n)⊗ \overline{D}(n) → \overline{D}(n) is the restriction of the Boolean algebra structure on 𝔽 ⊕ \overline{D}(n). This product may also be described as the restriction of the product on I_{V_{n}}.

One may consider the composite \overline{D}(n) ⊗ J_{λ} ↪ I_{V_{n}} ⊗ I_{V_{n}} → I_{V_{n}}, induced by inclusions of \overline{D}(n) and J_{λ} into I_{V_{n}} and where I_{V_{n}} ⊗ I_{V_{n}} → I_{V_{n}} is the product on I_{V_{n}}. It is clear that \{\overline{D}(n) ⊗ J_{λ}\}(𝔽^{n}) ≅ S_{λ}; this implies that the above map factors through some map J_{λ} ↪ I_{V_{n}}. Hence, the above discussion has proved:

Proposition 7.1.1. Suppose that S_{λ} is a simple GL_{n}-module. The multiplication on I_{V_{n}} restricts to a map \overline{D}(n) ⊗ J_{λ} → J_{λ}, which is a map of \overline{D}(n)-modules.

Corollary 7.1.2. Suppose that S_{λ} is a simple GL_{n}-module. There is a canonical surjection \overline{D}(n) ⊗ F_{λ} →→ J_{λ}.

Proof. There is certainly a unique non-trivial map given by the above constructions \overline{D}(n) ⊗ F_{λ} → J_{λ}, so that it is sufficient here to show that this map is surjective. Since the image is a sub-functor of J_{λ}, Theorem 6.0.1 shows that it is sufficient to show that the image is not nilpotent under the action of ˜∇_{n}. This is shown by using the description of this map given above and the fact that the product map induces an injection I_{V_{n}}(V) ⊗ I_{V_{n}}(W) ↪ I_{V_{n}}(V ⊕ W). Hence, the following composite is injective:

\overline{D}(n)(V) ⊗ F_{λ}(W) → (\overline{D}(n) ⊗ F_{λ})(V ⊕ W) → I_{V_{n}}(V ⊕ W).

This allows one to deduce that the image is not ˜∇_{n}-nilpotent, since \overline{D}(n) is not nilpotent. Essentially, one applies ( ˜∇_{n})^{t} with respect to a splitting of V as V′ ⊕ 𝔽^{t}.

Remark 7.1.3. The above results show that the structure of the co-Weyl functor J_{λ} induced from S_{λ} is intimately related to the structure of \overline{D}(n). This is further evidence that the functor \overline{D}(n) plays a fundamental rôle in the category F.

Acknowledgement

The author would like to thank Lionel Schwartz and Nick Kuhn for initiating him to the category F, for many conversations related to this subject area and also for their comments on a preliminary version of this paper^{1}. Finally, the author is grateful for the comments made by the referee and their careful reading of the text.

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LAGA, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France

   E-mail address: powell@math.univ-paris13.fr